Affine Toda solitons and systems of Calogero-Moser type

The solitons of affine Toda field theory are related to the spin-generalised Ruijsenaars-Schneider (or relativistic Calogero-Moser) models. This provides the sought after extension of the correspondence between the sine-Gordon solitons and the Ruijsenaars-Schneider model.     

Toda lattice hierarchy and conservation laws

Time evolutions of the Toda lattice hierarchies of Ueno and Takasaki are induced by Hamiltonians which are conservation laws for the original (one and two dimensional) Toda lattice obtained by Olive and Turok. Moreover these Hamiltonians for two dimensional Toda lattice hierarchy are also conserved quantities of the two component KP hierarchy in which that system is embedded. The one dimensional Toda lattice hierarchy is characterized by the bilinear relations, and a new version of the one dimensional Toda lattice hierarchy is constructed. Generalized Toda lattice hierarchies associated to all affine Lie algebras are presented.     

Toda lattice realization of integrable hierarchies

We present a new realization of scalar integrable hierarchies in terms of the Toda lattice hierarchy. In other words, we show on a large number of examples that an integrable hierarchy, defined by a pseudo-differential Lax operator, can be embedded in the Toda lattice hierarchy. Such a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld-Sokolov realization.     

From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.     

Remarks on the mass spectrum of two-dimensional Toda lattice of E8 type

A simple procedure for obtaining the mass spectrum of 2-dimensional Toda lattice of E₈ type is given.     

The Goryachev-Chaplygin top and the Toda lattice

The Goryachev-Chaplygin top is a rigid body rotating about a fixed point with principal moments of inertiaA, B, C satisfyingA=B=4C and with center of mass lying in the equatorial plane. The problem is algebraically completely integrable as a linear flow on a hyperelliptic Jacobian, only upon putting the principal angular momentum in the horizontal plane.This system admits asymptotic solutions with fractional powers int and depending on 4 degrees of freedom. As a consequence, the affine invariant surfaces of the Chaplygin top are double covers of the hyperelliptic Jacobian above, ramified along two translates of the theta divisor, touching in one point. This system is an instance of a (master) system of differential equations in 7 unknowns having 5 quadratic constants of motion; a careful analysis of this system reveals an intimate (rational) relationship with the 3-body periodic Toda lattice.The support of National Science Foundation grant No. DMS-8403136 is gratefully acknowledged. I thank Victor Guillemin for mentioning this problem     

Toda lattice hierarchy and generalized string equations

String equations of thep ^th generalized Kontsevich model and the compactifiedc=1 string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model atp=–1 does not coincide with thec=1 string theory at selfdual radius. A broader family of solutions of the Toda lattice hierarchy including these models is constructed, and shown to satisfy generalized string equations. The status of a variety ofc1 string models is discussed in this new framework.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

A systematic study of the Toda lattice

We apply the results of our recent investigation of integrable models to the particular case of the Toda lattice. We obtain the conserved quantities in closed form and derive the recursion relation which leads to their involution. We compare our Lax equation with the conventional and present the higher order Lax equations also in a closed form.     

A trihamiltonian extension of the Toda lattice

A new Poisson structure is defined on a subspace of the Kupershmidt algebra, isomorphic to the space H$\mathfrak{H}$ of n×n$n\times n$ Hermitian matrices. The new Poisson structure is of Lie–Poisson type with respect to the standard Lie bracket of H$\mathfrak{H}$. This Poisson structure (together with two already known ones, obtained through a r$r$-matrix technique) allows to construct an extension of the periodic Toda lattice with n$n$ particles that fits in a trihamiltonian recurrence scheme. Some explicit examples of the construction and of the first integrals found in this way are given.     

On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.     

Classical coulomb systems near a plane wall. I

The equilibrium structure of classical Coulomb systems bounded by a plane wall is studied near that wall. Several models are considered: the two-dimensional one-component plasma at a special value of the coupling constant (which makes the model exactly soluble), the two-dimensional and three-dimensional one-component and two-component plasmas in the weak-coupling limit (a Debye-Hückel type of approach is then used). Along a wall, the pair correlation functions decay only as an inverse power of the distancer, namely, asr ^–v for av-dimensional system (v=2,3). The one-body densities are also studied; the first BGY equation is used.     

Compatible Lie algebroids and the periodic Toda lattice

We give the set of maps from to the structure of a Poisson manifold endowed with a pair of compatible Lie algebroids. A suitable reduction process, of the Marsden–Ratiu type, yields a smaller manifold with the same geometrical properties as the original manifold. Moreover, is a bi-Hamiltonian manifold and the flows naturally defined on it are the periodic Toda flows.     

Master symmetries and R-matrices for the Toda lattice

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. These brackets may also be obtained by using r-matrices.     

HCIZ integral and 2D Toda lattice hierarchy

The expression of the large N Harish Chandra–Itzykson–Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large N HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.     

Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems

The probability density distributions for the ground states of certain model systems in quantum mechanics and for their classical counterparts are considered. It is shown that classical distributions are remarkably improved by incorporating into them the Heisenberg uncertainty relation between position and momentum. Even the crude form of this incorporation makes the agreement between classical and quantum distributions unexpectedly good, except for the small area, where classical momenta are large. It is demonstrated that the slight improvement of this form makes the classical distribution very similar to the quantum one in the whole space. The obtained results are much better than those from the WKB method. The paper is devoted to ground states, but the method applies to excited states too.     

Symmetry properties and bi-Hamiltonian structure of the Toda lattice

We carry out a systematic analysis of the Toda lattice equations developing a method which extends the symmetry approach formalism to discrete one-dimensional systems. We find a hereditary operator which admits a symplectic-implectic factorization. As a consequence of this property, we derive the Hamiltonian and the bi-Hamiltonian structure, together with the constants of motion and a set of infinitely-many commuting Lie-Bäcklund symmetries of the Toda chain.     

Classical and quantum mechanical systems of Toda-lattice type

Communications in Mathematical Physics - In a previous paper it was shown that certain Schrödinger operatorsH=Δ − V onR ℓ such as the Hamiltonians for the quantized...